Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability

Synchronization transitions are investigated in small-world neuronal networks that are locally modeled by the Rulkov map with additive spatiotemporal noise. In particular, we investigate the impact of different information transmission delays and rewiring probability. We show that short delays induce zigzag fronts of excitations, whereas intermediate delays can further detriment synchrony in the network due to a dynamic clustering anti-phase synchronization transition. Detailed investigations reveal, however, that for longer delay lengths the synchrony of excitations in the network can again be enhanced due to the emergence of in-phase synchronization. In addition, we show that an appropriate small-world topology can restore synchronized behavior provided information transmission delays are either short or long. On the other hand, within the intermediate delay region, which is characterized by anti-phase synchronization and clustering, differences in the network topology do not notably affect the synchrony of neuronal activity. Copyright c © EPLA, 2008 Introduction. – The complex dynamics of networks of coupled neurons is a central topic in theoretical neuroscience [1]. Within the array of possible dynamical behaviors, wave formation and synchronization [2] seem to be very important for the efficient processing and transmission of information across the nervous system. The brain is a highly distributed multitasking system in which numerous operations are executed in a parallel fashion, yet it lacks a main coordination center. One of the coordinating mechanisms appears to be the synchronization of neuronal activity via phase locking [3] of self-generated oscillations. In the past decade many theoretical and experimental works have been performed with the goal to analyze wave formation and synchronization in a vast variety of different systems, including complex networks [4]. Paths to synchronization on complex networks have been investigated in [5], whereas universalities in the synchronization of weighted random networks (a)E-mail: [email protected] were reported in [6]. More specifically related to neuronal systems, subthreshold stimulus-aided synchronization and wave formation on a square lattice of noisy neurons have been investigated by means of the Rulkov map [7]. Moreover, Sato et al. studied the effects of different widths of the action potential on synchronization phenomena of coupled neurons [8]. Complex spatiotemporal behaviors, including in-phase and anti-phase synchronization as well as various wave formation patterns, have been observed in a ring network of discrete bursting oscillators [9]. Pattern formation and firing synchronization have also been studied in a network of discrete neurons as the coupling strength was varied [10]. According to the Watts-Strogatz small-world network [11], neurons of the brain are coupled mainly locally, but in addition, are also connected through sparse long-range connections linking physically distant units. Convincing evidences have been presented which support the idea that small-world networks provide a powerful and versatile tool, leading us towards understanding